Final answer:
To rewrite the quadratic function y=3x²+30x+65 in vertex form, we complete the square to find the vertex form as y=3(x+5)²-10, which corresponds to option B.
Step-by-step explanation:
To rewrite the quadratic function y=3x²+30x+65 in vertex form, we start by completing the square. The vertex form of a quadratic function is y=a(x-h)²+k where (h,k) is the vertex of the parabola.
First, we factor out the coefficient of the x² term from the x terms:
y = 3(x² + 10x) + 65
Then, we find the value to complete the square for the x terms. This value is (b/2a)² where a is the coefficient of x² and b is the coefficient of x. Thus, we have (10/2)² = 25.
Now, we add and subtract this value inside the parenthesis:
y = 3(x² + 10x + 25 - 25) + 65
This simplifies to:
y = 3((x + 5)² - 25) + 65
Finally, we distribute the 3 and combine like terms:
y = 3(x + 5)² - 75 + 65
y = 3(x + 5)² - 10
The correct option that represents the quadratic function in vertex form is B. y=3(x+5)²-10.